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BEGIN:VEVENT
SUMMARY:Petter Braenden (KTH Royal Institute of Technology)
DTSTART:20211018T132000Z
DTEND:20211018T141000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/1
DESCRIPTION:Title: Stable polynomials and related families of polynomials\nby
Petter Braenden (KTH Royal Institute of Technology) as part of CMO- Real P
olynomials: Counting and Stability\n\n\nAbstract\nI will give a panoramic
talk on stable polynomials and related families of polynomials\, such as h
yperbolic and Lorentzian polynomials. Over the past two decades stable pol
ynomials and their relatives have been applied in different areas such as
optimization\, real algebraic geometry\, combinatorics\, statistical mecha
nics\, quantum mechanics and computer science. I will review some remarkab
le properties of this class of polynomials as well as point to application
s. I will also talk about a recent generalization called Lorentzian polyno
mials.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frédéric Bihan (Universite Savoie Mont Blanc)
DTSTART:20211018T143000Z
DTEND:20211018T152000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/2
DESCRIPTION:Title: Fewnomial bounds and multivariate generalisations of Descartes
’ rule of signs\nby Frédéric Bihan (Universite Savoie Mont Blanc)
as part of CMO- Real Polynomials: Counting and Stability\n\n\nAbstract\nIn
1980\, A. Khovanskii gave a bound for the number of non-degenerate positi
ve solutions of any square sparse polynomial system. His bound depends onl
y on the number of monomials of the system and is smaller than all classic
al bounds (Bézout or mixed volume bounds) when the number of monomials is
small. Such bounds are called fewnomial bounds. In the univariate case\,
the classical Descartes’ rule of signs\, going back from 1637\, produces
a bound for the number of positive roots which takes care of the signs of
the coefficients\, which is sharp and which implies a sharp fewnomial bou
nd. In this talk\, I will describe several improvements of Khovanskii boun
d\, which in some cases provide sharp fewnomial bounds. I will also descri
be recent multivariate generalisations of Descartes’ rule of signs. This
talk is mainly based on joint works with several collaborators including
A. Dickenstein\, B. El Hilany\, J. Forsgard\, M. Rojas and F. Sottile.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thorsten Theobald (Goethe-Universität Frankfurt/Main)
DTSTART:20211018T154000Z
DTEND:20211018T163000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/3
DESCRIPTION:Title: Conic stability of polynomials\, imaginary projections and spec
trahedra\nby Thorsten Theobald (Goethe-Universität Frankfurt/Main) as
part of CMO- Real Polynomials: Counting and Stability\n\n\nAbstract\nA mu
ltivariate polynomial $p$ in ${\\mathbb C}[z_1\, \\ldots\, z_n]$\nis calle
d stable if every root $z$ has at least one\ncomponent $z_j$ with imaginar
y part $\\le 0$. In this\nexpository talk\, we discuss the naturally gener
alized\nviewpoint of conic stability. Its origin is in the\nstudy of imagi
nary projections\, and the usual stability\nrefers to the specific polyhed
ral cone ${\\mathbb R}_+^n$.\n\nAs a prominent case\, we consider conic st
ability with\nrespect to the positive semidefinite cone ("psd stability").
\nCriteria for psd stability are tightly linked to the\ncontainment proble
m for spectrahedra\, to positive maps\nand to determinantal representation
s.\n\nThe own results in this talk are based on various joint\nworks with
Giulia Codenotti\, Papri Dey\, Stephan Gardoll\,\nThorsten Jörgens\, Mahs
a Sayyary and Timo de Wolff.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Shapiro (University of Stockholm)
DTSTART:20211019T130000Z
DTEND:20211019T134000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/4
DESCRIPTION:Title: Return of the plane evolute\nby Boris Shapiro (University o
f Stockholm) as part of CMO- Real Polynomials: Counting and Stability\n\n\
nAbstract\nWe consider the evolutes of plane real-algebraic curves and dis
cuss some of their complex and real-algebraic properties. In particular\,
for a given degree d ≥ 2\, we provide lower bounds for the following fou
r numerical invariants: 1) the maximal number of times a real line can int
ersect the evolute of a real-algebraic curve of degree d\; 2) the maximal
number of real cusps which can occur on the evolute of a real-algebraic cu
rve of degree d\; 3) the maximal number of (cru)nodes which can occur on t
he dual curve to the evolute of a real-algebraic curve of degree d\; 4) th
e maximal number of (cru)nodes which can occur on the evolute of a real-al
gebraic curve of degree d (joint with R.Piene and C.Riener).\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristhian Garay López (CIMAT)
DTSTART:20211019T140000Z
DTEND:20211019T144000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/5
DESCRIPTION:Title: Inflection polynomials of linear series on superelliptic curves
\nby Cristhian Garay López (CIMAT) as part of CMO- Real Polynomials:
Counting and Stability\n\n\nAbstract\nWe explore the inflectionary behavio
r of linear series on families of marked superelliptic curves (i.e.\, cycl
ic covers of $\\mathbb{P}^1$). The inflection of these linear series supp
orted away from the superelliptic ramification locus is parameterized by t
he inflection polynomials\, a certain family of polynomials generalizing
the division polynomials (which are used to compute the torsion points of
elliptic curves). These polynomials are remarkable since their properties
reflect aspects of the underlying family of superelliptic curves. We also
obtain inflectionary varieties\, which describe the global behaviour of th
e inflection points on the family.\n\nIn this talk we will introduce these
inflection polynomials and some of their properties. Although this story
is valid over fields of arbitrary characteristic\, we will focus on the re
al case. \nWe report on joint work with Ethan Cotterill\, Ignacio Darago\,
Changho Han\, and\nTony Shaska.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mareike Dressler (UC San Diego)
DTSTART:20211019T150000Z
DTEND:20211019T154000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/6
DESCRIPTION:Title: Real zeros of sums of nonnegative circuit polynomials\nby M
areike Dressler (UC San Diego) as part of CMO- Real Polynomials: Counting
and Stability\n\n\nAbstract\nUnderstanding the real zeros of polynomials i
s a research subject of intrinsic interest with a long and rich history an
d is especially useful for polynomials with specific properties like nonne
gativity. In this talk\, I provide a complete and explicit characterizatio
n of the real zeros of both homogeneous and inhomogeneous sums of nonnegat
ive circuit (SONC) polynomials\, a recent certificate for nonnegative poly
nomials independent of sums of squares. As an interesting consequence\, I
show that the supremum of the number of zeros of all homogeneous n-variat
e polynomials of degree 2d in the SONC cone can be determined exactly. No
te that in strong contrast\, the determination of this number for both the
nonnegativity cone and the cone of sums of squares for general n and d
is still an open question.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cynthia Vinzant (University of Washington)
DTSTART:20211019T160000Z
DTEND:20211019T164000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/7
DESCRIPTION:Title: Log-concavity and applications to approximate counting and samp
ling in matroids\nby Cynthia Vinzant (University of Washington) as par
t of CMO- Real Polynomials: Counting and Stability\n\n\nAbstract\nMatroids
are combinatorial objects designed to capture independence relations on c
ollections of objects\, such as linear independence of vectors in a vector
space or cyclic independence of edges in a graph. Recent work by several
independent authors shows that the multivariate basis-generating polynomia
l of a matroid is log-concave as a function on the positive orthant. In th
is talk\, I will describe some of the underlying combinatorial and geometr
ic structure of such log-concave polynomials and applications to the probl
ems of approximately counting and approximately sampling the bases of a ma
troid. This is based on joint work with Nima Anari\, Kuikui Liu\, and Shay
an Oveis Gharan.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claus Scheiderer (Univ-Konstanz Germany)
DTSTART:20211020T130000Z
DTEND:20211020T134000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/8
DESCRIPTION:Title: Low-complexity semidefinite representation of convex hulls of c
urves\nby Claus Scheiderer (Univ-Konstanz Germany) as part of CMO- Rea
l Polynomials: Counting and Stability\n\n\nAbstract\nMatroids are combinat
orial objects designed to capture independence relations on collections of
objects\, such as linear independence of vectors in a vector space or cyc
lic independence of edges in a graph. Recent work by several independent a
uthors shows that the multivariate basis-generating polynomial of a matroi
d is log-concave as a function on the positive orthant. In this talk\, I w
ill describe some of the underlying combinatorial and geometric structure
of such log-concave polynomials and applications to the problems of approx
imately counting and approximately sampling the bases of a matroid. This i
s based on joint work with Nima Anari\, Kuikui Liu\, and Shayan Oveis Ghar
an.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:El Hilany Boulos (TU Braunschweig)
DTSTART:20211020T140000Z
DTEND:20211020T144000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/9
DESCRIPTION:Title: A polyhedral description for the non-properness set of a polyno
mial map\nby El Hilany Boulos (TU Braunschweig) as part of CMO- Real P
olynomials: Counting and Stability\n\n\nAbstract\nLet $K$ be the field of
either real or complex numbers\, and let $S_f$ denote the set of points in
$K^n$ at which a polynomial map $f: K^n\\to K^n$ is not proper.\nJelonek
proved that $S_f$ is an algebraic hypersurface in the complex case and sem
i-algebraic in the real case. He furthermore showed that $S_f$ is ruled by
polynomial\ncurves\, and provided a method for computing its equations fo
r complex maps. However\, such methods do not extend to real polynomial ma
ps.\n\nIn this talk\, I will establish a description of $S_f$ for\na large
family of non-proper polynomial maps f using their Newton polytopes. I wi
ll furthermore highlight the interplay between the combinatorics of the po
lytopes and the topology of $S_f$. The resulting method computes $S_f$ for
complex polynomial maps as well as the real ones. As an application\, som
e of\nJelonek's results are recovered. \n\nI will furthermore report on a
joint work with Elias Tsigaridas in which we provided a more elaborate me
thod for computing the non-properness set for degenerate real polynomial m
aps on the plane.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mario Kummer (TU Berlin Germany)
DTSTART:20211020T150000Z
DTEND:20211020T154000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/10
DESCRIPTION:Title: Matroids with the half-plane property and related concepts
\nby Mario Kummer (TU Berlin Germany) as part of CMO- Real Polynomials: Co
unting and Stability\n\n\nAbstract\nWe will study several properties of ba
ses generating polynomials of matroids that are related to stability. This
includes the half-plane property or determinantal representability. We wi
ll further present a classification of all matroids on up to eight element
s whose bases generating polynomial is stable. This is joint work with Bü
şra Sert.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Papri Day (University of Missouri)
DTSTART:20211020T160000Z
DTEND:20211020T164000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/11
DESCRIPTION:Title: Real Degeneracy Loci of Matrices\, and Hyperbolicity cones of
Real Polynomials\nby Papri Day (University of Missouri) as part of CMO
- Real Polynomials: Counting and Stability\n\n\nAbstract\nThis talk has tw
o parts. In the first part\, I shall talk about real degeneracy loci of ma
trices and its correspondence with symmetroids. Let $\\mathcal{A}:=\\{A_1
\\dots\,A_{m+1}\\}$ be a collection of linear operators on ${\\mathbb R}^{
m}$. The degeneracy locus (DL) of $\\mathcal{A}$ is defined as the set of
the points $x$ for which rank$([A_1x\\dots A_{m+1}x])\\leq m-1$. We show t
hat the DL is an $m-3$ dimensional sub-scheme of degree ${m+1 \\choose 2}$
in ${\\mathbb P}^{m-1}({\\mathbb C})$. In particular\, when $m=3$\, the D
L consists of six rational points in ${\\mathbb P}^{2}({\\mathbb R})$ with
quadrilateral configuration if and only if $A_{i}\,i=1\\dots\,4$ are in t
he linear span of four fixed rank-one operators. Moreover\, we show that i
f $A_{i}\,i=1\\dots\,m+1$ are in the linear span of $m+1$ fixed rank-one m
atrices\, the DL of $m+1$ matrices satisfies generalized Desargues configu
ration\, and it has correspondence with a Special type of symmetroid\, cal
l it Sylvester symmetroid. This part is based on joint work with Dan Edidi
n.\n\nIn the second part\, I shall focus on the hyperbolicity cones of the
elementary symmetric polynomials and real polynomials which define symmet
roids (work in progress).\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahsa Sayyary Namin (Goethe University Frankfurt)
DTSTART:20211021T130000Z
DTEND:20211021T134000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/12
DESCRIPTION:Title: Imaginary Projections: Complex Versus Real Coefficients\nb
y Mahsa Sayyary Namin (Goethe University Frankfurt) as part of CMO- Real P
olynomials: Counting and Stability\n\n\nAbstract\nGiven a complex multivar
iate polynomial \n${p\\in\\mathbb{C}[z_1\,\\ldots\,z_n]}$\, the imaginary
projection \n$\\mathcal{I}(p)$ of $p$ is defined as the projection of the
variety \n$\\mathcal{V}(p)$ onto its imaginary part. We give a full \nchar
acterization of the imaginary projections of conic sections with \ncomplex
coefficients\, which generalizes a classification for the case of \nreal
conics. More precisely\, given a bivariate complex polynomial \n$p\\in\\ma
thbb{C}[z_1\,z_2]$ of total degree two\, we describe the number \nand the
boundedness of the components in the complement of \n$\\mathcal{I}(p)$ as
well as their boundary curves and the spectrahedral \nstructure of the com
ponents. We further study the imaginary projections \nof some families of
higher degree complex polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mauricio Velasco (Universidad de los Andes)
DTSTART:20211021T140000Z
DTEND:20211021T144000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/13
DESCRIPTION:Title: Harmonic hierarchies for polynomial optimization\nby Mauri
cio Velasco (Universidad de los Andes) as part of CMO- Real Polynomials: C
ounting and Stability\n\n\nAbstract\nThe cone of nonnegative forms of a gi
ven degree is a convex set of remarkable beauty and usefulness.\nIn this t
alk we will discuss some recent ideas for approximating this set through p
olyhedra and spectrahedra. We call the resulting approximations harmonic h
ierarchies since they arise naturally from harmonic analysis on spheres (o
r equivalently from the representation theory of $SO(n)$). We will describ
e theoretical results leading to sharp estimates for the quality of these
approximations and will also show a brief demo of our Julia implementation
of harmonic hierarchies. These results are joint work with Sergio Cristan
cho (UniAndes).\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josephine Yu (Georgia Institute of Technology)
DTSTART:20211021T150000Z
DTEND:20211021T154000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/14
DESCRIPTION:Title: Positively Hyperbolic Varieties\, Tropicalization\, and Positr
oids\nby Josephine Yu (Georgia Institute of Technology) as part of CMO
- Real Polynomials: Counting and Stability\n\n\nAbstract\nWe will discuss
a generalization of stable polynomials to complex algebraic varieties of c
odimension larger than one and study their combinatorial structure using t
ropical geometry. We show that their tropicalization are closely related t
o type-A braid arrangements and positroids (matroid arising from the nonne
gative part of the Grassmannian) and that their Chow polytopes are general
ized permutohedra. This is based on joint work with Felipe Rincón and Cyn
thia Vinzant.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Máté László Telek (University of Copenhagen)
DTSTART:20211021T155000Z
DTEND:20211021T160500Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/15
DESCRIPTION:Title: On generalizing Descartes' rule of signs to hypersurfaces\
nby Máté László Telek (University of Copenhagen) as part of CMO- Real
Polynomials: Counting and Stability\n\n\nAbstract\nWe provide upper bounds
on the number of connected components of the complement of a hypersurface
in the positive orthant and phrase our results as partial generalizations
of the classical Descartes’ rule of signs to multivariate polynomials (
with real exponents). In particular\, we give conditions based on the geom
etrical configuration of the exponents and the sign of the coefficients th
at guarantee that the number of connected components of the complement of
the hypersurface where the defining polynomial attains a negative value is
at most one or two. Furthermore\, we briefly present an application for c
hemical reaction networks that motivated us to consider this problem.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abeer Al Ahmadieh (University of Washington)
DTSTART:20211021T160500Z
DTEND:20211021T162000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/16
DESCRIPTION:Title: Determinantal Representations and the Image of the Principal M
inor Map\nby Abeer Al Ahmadieh (University of Washington) as part of C
MO- Real Polynomials: Counting and Stability\n\n\nAbstract\nThe principal
minor map takes an $n$ by $n$ square matrix to the length-$2^n$ vector o
f its principal minors. A basic question is to give necessary and sufficie
nt conditions that characterize the image of various spaces of matrices un
der this map. In this talk I will describe the image of the space of compl
ex matrices using a characterization of determinantal representations of m
ultiaffine polynomials\, based on the factorization of their Rayleigh diff
erences. Using these techniques I will give equations and inequalities cha
racterizing the images of the spaces of real and complex symmetric\, Hermi
tian\, and general complex matrices. For complex symmetric matrices this r
ecovers a result of Oeding from $2011$. This is based on a joint work with
Cynthia Vinzant.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cédric Le Texier (Oslo University)
DTSTART:20211021T162000Z
DTEND:20211021T163500Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/17
DESCRIPTION:Title: Hyperbolic plane curves near the non-singular tropical limit\nby Cédric Le Texier (Oslo University) as part of CMO- Real Polynomial
s: Counting and Stability\n\n\nAbstract\nWe develop tools of real tropical
intersection theory in order to determine necessary and sufficient condit
ions for real algebraic curves near the non-singular tropical limit to be
hyperbolic with respect to a point\, in terms of real phase structure and
twisted edges on a tropical curve\, generalising Speyer's classification o
f stable curves near the tropical limit.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josué Tonelli-Cueto (Inria Paris & IMJ-PRG)
DTSTART:20211021T163500Z
DTEND:20211021T165000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/18
DESCRIPTION:Title: Metric restrictions on the number of real zeros\nby Josué
Tonelli-Cueto (Inria Paris & IMJ-PRG) as part of CMO- Real Polynomials: C
ounting and Stability\n\n\nAbstract\nA well-known fact in real algebraic g
eometry is that crossing the discriminant changes the number of real zeros
. However\, can the size of a discriminant chamber influence the number of
zeros of the polynomial systems in it? In this talk\, we show some novel
results showing that this is the case. More concretely\, we show that we c
an bound the number of real zeros in terms of the logarithm of the inverse
distance to the discriminant—also known as the condition number—. We
also demonstrate that this bound has important consequences regarding rand
om real polynomial systems. This is joint work with Elias Tsigaridas.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Khazhgali Kozhasov (Universität Osnabrück)
DTSTART:20211022T130000Z
DTEND:20211022T134000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/19
DESCRIPTION:Title: The many faces of polynomial capacity\nby Khazhgali Kozhas
ov (Universität Osnabrück) as part of CMO- Real Polynomials: Counting an
d Stability\n\n\nAbstract\nThe capacity of a polynomial p with non-negativ
e coefficients is a certain function on its support that interpolates betw
een coefficients of p and its value at the vector (1\,...\,1). This concep
t has a lot of remarkable applications including bounds on the mixed volum
e of convex bodies and bounds on some combinatorial quantities like the nu
mber of matchings in bipartite graphs. I will discuss relation of capacity
to relative entropy of measures as well as its appearances in the theory
of non-negative polynomials and in the theory of A-discriminants. The talk
is based on a joint work in progress with Jonathan Leake and Timo de Wolf
f.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simone Naldi (Université de Limoges)
DTSTART:20211022T140000Z
DTEND:20211022T144000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/20
DESCRIPTION:Title: Spectrahedral representations of hyperbolic plane curves\n
by Simone Naldi (Université de Limoges) as part of CMO- Real Polynomials:
Counting and Stability\n\n\nAbstract\nA key question in the theory of hyp
erbolic polynomials is how to test hyperbolicity. This is classically done
by computing a symmetric determinantal representation of the given polyno
mial. In the case of curves this is always possible\, whereas in high dime
nsion one should look at such representations for multiples of the given p
olynomial (according to the Generalized Lax Conjecture). In the talk I wil
l discuss a recent variant of the classical Dixon method\, for the computa
tion of spectrahedral representations of curves. The talk is based on a re
cent work joint with Mario Kummer and Daniel Plaumann.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:J. Maurice Rojas (exas A & M University)
DTSTART:20211022T150000Z
DTEND:20211022T154000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/21
DESCRIPTION:Title: Counting Pieces of Real Near-Circuit Hypersurfaces Faster\
nby J. Maurice Rojas (exas A & M University) as part of CMO- Real Polynomi
als: Counting and Stability\n\n\nAbstract\nRandomization has proved instru
mental in efficiently solving\ngeometric problems where the best determini
stic methods are impractical.\nAn important recent example is a recent sin
gly exponential algorithm of\nBurgisser\, Cucker\, and Tonelli-Cueto for c
omputing the homology of real\nalgebraic sets for ``most'' inputs. We appr
oach an analogous speed-up in a\ndifferent direction: Computing the isotop
y type of real zero sets\ndefined by certain n-variate sparse polynomials
of degree d with\ncoefficients of maximum bit-length h. We show how\,\nfor
``most'' inputs\, we can compute the number of connected components\nof t
he positive zero set in time $(h log d)^O(n)$\, whereas the fastest\nprevi
ous algorithms had complexity $(hd)^{O(n)}$. A key tool is a new\nway to m
etrically approximate certain A-discriminant varieties. We'll aslo\nsee ho
w reducing the dependence on the number of variables n is related\nto diop
hantine approximation.\n\nParts of this work are joint with Frederic Bihan
\, Jens Forsgard\, Mounir\nNisse\, Kaitlyn Phillipson\, and Lisa Soule.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucia Lopez de Medrano (Universidad Nacional Autonoma de Mexico)
DTSTART:20211022T160000Z
DTEND:20211022T164000Z
DTSTAMP:20260404T041640Z
UID:CMO-21w5117/22
DESCRIPTION:Title: On maximally inflected hyperbolic curves\nby Lucia Lopez d
e Medrano (Universidad Nacional Autonoma de Mexico) as part of CMO- Real P
olynomials: Counting and Stability\n\n\nAbstract\nIn this talk we will foc
us on the distribution of real inflection points among the ovals of a real
non-singular hyperbolic curve of even degree. Using Hilbert’s method we
show that for any integers $d$ and $r$ such that $4 ≤ r ≤ 2d^2 −2d$
\, there is a non-singular hyperbolic curve of degree $2d$ in $\\mathbb R^
2$ with exactly $r$ line segments in the boundary of its convex hull. We a
lso give a complete classification of possible distributions of inflection
points among the ovals of a maximally inflected non-singular hyperbolic c
urve of degree 6. This is a joint work with Aubin Arroyo and Erwan Brugall
é.\n
LOCATION:https://stable.researchseminars.org/talk/CMO-21w5117/22/
END:VEVENT
END:VCALENDAR